Difference Between Transitive Property and Substitution Property

Transitive Property vs Substitution Property

The substitution property is used for values or variables that represent numbers. The substitution property of equality states that for any numbers a and b, if a = b, then a may be replaced with b. Therefore, if a=b, then we can change any ‘a’ to a ‘b’ or any ‘b’ to an ‘a’.

For example, if it is given that x=6, then we can solve the expression (x+4)/5 by substituting the value of x. By substituting 5 for x in the above expression; (6+4)/5 = 2. Essentially, any two values can be substituted for one another, if and only if, they are equal to each other.

There is a substitution property defined in geometry. According to this substitution property definition, if two geometric objects (it can be two angles, segments, triangles, or whatever) are congruent, then these two geometric objects can be replaced with one other in a statement involving one of them.

Transitive property is a more formal definition, which is defined on binary relations.  A relation R from the set A to the set B is a set of ordered pairs, if A and B are equal, we say that the relation is a binary relation on A. Transitive property is one out of the properties (Reflexive, Symmetric, Transitive) used to define equivalence relations.

A relation R is transitive, if and only if, x is related by R to y, and y is related by R to z, then x is related by R to z. Symbolically, a transitive property can be defined as follows. Let  a, b and c belonging to a set A, a binary relation ‘~’ has the transitive property defined by,If a ~ b and b ~ c, then that implies a ~ c.

For an example, “being greater than” is a transitive relation. If a, b and c are any real numbers such that, a is greater than b, and b is greater than c, then it is a logical consequence that a is greater than c. “Being taller” is also a transitive relation. If Kate is taller than Mary, and Mary is taller than Jenney, it implies that Kate is taller than Jenney.

We cannot apply transitive relation criteria on all binary relations. For example, if Bill is John’s father and John is Fred’s father, which does not imply that Bill is Fred’s father. Similarly, “likes” is non transitive property. If Wilson likes Henry and Henry likes David, that does not imply that Wilson likes David. Hence, it is not a transitive relation.

In geometry, Transitive Property (for three segments or angles) is defined as follows:

If two segments (or angles) are each congruent with a third segment (or angle), then they are congruent with each other.

The transitive property of equality is defined as follows. Let a, b and c are any three elements in set A, such that a=b and b=c, then a=c. This looks similar to substitution property, which can be considered replacing b with c in the equation a=b. However, these two properties are not the same.